Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. the population of a town can be modeled using the formula $p = 20,000e^{0.02t}$, where $t$ is the number of years after 2012 and $p$ is the towns population. which of the following equations can be used to find the number of years after 2012 that the population will double to 40,000? $t = \frac{\log 2}{0.02}$ $t = \frac{2}{0.02e}$ $t = \frac{\ln 2}{0.02}$ $t = \frac{\ln 20,000}{0.02}$

Explanation:

Step1: Set P to 40000

We know the population formula is \( P = 20000e^{0.02t} \). When the population doubles to 40000, we substitute \( P = 40000 \) into the formula:
\( 40000 = 20000e^{0.02t} \)

Step2: Simplify the equation

Divide both sides by 20000:
\( \frac{40000}{20000} = e^{0.02t} \)
Simplifying the left side gives \( 2 = e^{0.02t} \)

Step3: Take natural log of both sides

To solve for \( t \), take the natural logarithm (ln) of both sides. Recall that \( \ln(e^x) = x \), so:
\( \ln(2) = \ln(e^{0.02t}) \)
\( \ln(2) = 0.02t \)

Step4: Solve for t

Divide both sides by 0.02 to isolate \( t \):
\( t = \frac{\ln 2}{0.02} \)

Answer:

\( t = \frac{\ln 2}{0.02} \) (the third option)